Optimality Conditions and Uniqueness of the Solution in Nonsmooth Optimization

Résumé

Over the last several decades, tremendous efforts of researchers have been concentrated on the study of optimization problems with nonsmooth data. This is because most of practical models arisen in engineering, economics and other applied sciences involve nondifferentiable functions. Among theoretical topics of optimization, necessary conditions and sufficient conditions for optimal solutions play a central role. They provide a tool to select the good solutions and eliminate the bad ones. The uniqueness of solutions is another important topic as it often guarantees the convergence of algorithms for finding optimal solutions. These two relevant themes are main focus of our thesis. From Fermat's theorem, the first mathematically rigorous result on optimality conditions proved in the 17th century, it is commonly recognized the crucial role of the notion of derivative in the study of optimality conditions. Therefore, to deal with nonsmooth problems, a wide range of generalized derivatives has been introduced, which replace the nonexistent classical derivative. Extensive search for generalized derivatives suitable for nonsmooth optimization problems in recent years is the main inspiration of our work. In this thesis, we use three kinds of known generalized derivatives: the first and second-order approximations introduced in [63] and [1], the Hadamard first and second-order (upper) directional derivatives introduced in [95] and the approximate Jacobian and Hessian proposed in [56] to study single-valued optimization problems. However, we shall also discuss results using other generalized derivatives. One of the reasons for choosing these notions of derivatives is that even discontinuous mappings may have second-order approximations and/or the Hadamard derivative, and continuous mappings always have approximate Jacobians. Regular properties like local Lipschitz continuity are not needed.

la date de réponse	2007
langue originale	Anglais
Superviseur	Phan Quoc Khanh (Promoteur), Jean-Jacques STRODIOT (Jury), Jean-Paul Penot (Jury) & Dinh The Luc (Jury)